Mechanical Engineering Principles

SIMPLY SUPPORTED BEAMS 63

1.2 m

12 kN 400 N 20 kN

1.3 m 1.5 m

PQ

RP RQ

Figure 5.14

(a) At equilibrium,

RP+RQ= 12 + 0. 4 + 20

= 32 .4kN ( 1 )

Taking moments about P: clockwise mo-

ments=anticlockwise moments i.e.

( 12 × 1. 2 ) + ( 0. 4 × 2. 5 )

+ ( 20 × 3. 5 )

}

=(RQ× 5. 0 )

14. 4 + 1. 0 + 70. 0 = 5. 0 RQ

from which,

RQ=

85. 4

5. 0

= 17 .08 kN

From equation (1),

RP= 32. 4 −RQ= 32. 4 − 17. 08

= 15 .32 kN

(b) For the reactions of the supports to be equal,

RP=RQ=

32. 4

2

= 16 .2kN

Let the 12 kN load be at a distancedmetres

fromP (instead of at 1.2 m fromP). Taking

moments about pointPgives:

( 12 ×d)+( 0. 4 × 2. 5 )+( 20 × 3. 5 )= 5. 0 RQ

i.e. 12 d+ 1. 0 + 70. 0 = 5. 0 × 16. 2

and 12 d= 81. 0 − 71. 0

from which, d=

10. 0

12

= 0 .833 m

Hence the 12 kN load needs to be moved to a

position 833 mm fromPfor the reactions of the

supports to be equal(i.e. 367 mm to the left of its

original position).

Problem 10. A uniform steel girderABis

6.0 m long and has a mass equivalent to

4.0 kN acting at its centre. The girder rests

on two supports atCandBas shown in

Figure 5.15. A point load of 20.0 kN is

attached to the beam as shown. Determine

the value of the forceFthat causes the beam

to just lift off the supportB.

F

3.0 m 1.0 m

4.0 kN 20.0 kN

2.5 m

A C B

RC RB

Figure 5.15

At equilibrium,RC+RB=F+ 4. 0 + 20 .0.

When the beam is just lifting off of the supportB,

thenRB=0, henceRC=(F+ 24. 0 )kN.

Taking moments aboutA:

Clockwise moments=anticlockwise moments

i.e. ( 4. 0 × 3. 0 )+( 5. 0 × 20. 0 )=(RC× 2. 5 )

+(RB× 6. 0 )

i.e. 12. 0 + 100. 0 =(F+ 24. 0 )× 2. 5

+( 0 )

i.e.

112. 0

2. 5

=(F+ 24. 0 )

from which, F= 44. 8 − 24. 0

= 20 .8kN

i.e.the value of forceFwhich causes the beam

to just lift off the supportBis 20.8 kN.

Now try the following exercise

Exercise 27 Further problems on simply

supported beams having

point loads

1. Calculate the force RA and distance d
   for the beam shown in Figure 5.16. The
   mass of the beam should be neglected and
   equilibrium conditions assumed.
   [2 kN, 24 mm]